Okay, so we can boil it down to the fact that you can't send normal Text Messages from your Mac.
Reply to Side Note 1: Message uses iCloud to back up your messages. Not for sending Text Messages, that's what is meant by Messages using iCloud. You can say, however, that in order for your Mac to send text messages from your Mac, your iPhone and Mac must be under the same Apple ID, therefore also iCloud. But I don't think it's iCloud because it works fine on my Mac.
Reply to Side Note 2: Possibly. Montery is not the latest MacOS version, but it should still work.
But the problem is Text Forwarding. When you send a green text message from your Mac, it is actually sent from your iPhone, and it is the connection that needs to be fixed.
Try this, on your iPhone go to:
Settings → Messages → Text Message Forwarding, disable your Mac.
On your Mac, open Messages, choose Settings/Preferences, tap iMessages, and Sign Out. So your Mac is not receiving messages. Then restart your Mac, and your iPhone, again.
Open Settings in Messages on your Mac again and sign in with your Apple ID. Once you're logged in, notice if your iPhone asks if you want to enable Text Forwarding to your Mac. If it doesn't ask, go in and activate your Mac in Settings → Messages → Text Message Forwarding